A Statistical Field Approach to Capital Accumulation
Pierre Gosselin, Aïleen Lotz, Marc Wambst
Abstract: This paper presents a model of capital accumulation for a large number of heterogenous producer-consumers in an exchange space in which interactions depend on agents’ positions. Each agent is described by his production, consumption, stock of capital, as well as the position he occupies in this abstract space.
Each agent produces one differentiated good whose price is fixed by market clearing conditions. Production functions are Cobb-Douglas, and capital stocks follow the standard capital accumulation dynamic equation. Agents consume all goods but have a preference for goods produced by their closest neighbors.
Agents in the exchange space are subject both to attractive and repulsive forces. Exchanges drive agents closer, but beyond a certain level of proximity, agents will tend to crowd out more distant agents.
The present model uses a formalism based on statistical field theory developed earlier by the authors. This approach allows the analytical treatment of economic models with an arbitrary number of agents, while preserving the system’s interactions and complexity at the individual level.
Our results show that the dynamics of capital accumulation and agents’ position in the exchange space are correlated. Interactions in the exchange space induce several phases of the system.
A first phase appears when attractive forces are limited. In this phase, an initial central position in the exchange space favors capital accumulation in average and leads to a higher level of capital, while agents far from the center will experience a slower accumulation process. A high level of initial capital drives agents towards a central position, i.e. improve the terms of their exchanges: they experience a higher demand and higher prices for their product. As usual, high capital productivity favors capital accumulation, while higher rates of capital depreciation reduce capital stock.
In a second phase, attractive forces are predominant. The previous results remain, but an additional threshold effect appears. Even though no restriction was imposed initially on the system, two types of agents emerge, depending on their initial stock of capital. One type of agents will remain above the capital threshold and occupy and benefit from a central position. The other type will remain below the threshold, will not be able to break it and will remain at the periphery of the exchange space. In this phase, capital distribution is less homogenous than in the first phase.
Key words: Path Integrals, Statistical Field Theory, Phase Transition, Capital Accumulation, Exchange Space, Multi-Agent Model, Interaction Agents.
JEL Classification: C02, C60, E00, E1.
Financial Interactions and Collective States: Banks, Investors and Firms
Pierre Gosselin, Aïleen Lotz
Abstract: In a previous paper, we applied a field formalism to analyze capital allocation and accumulation within a microeconomic framework of investors and firms. The financial connections were modeled by a field of stakes, representing the links between agents. We showed that the resulting collective states were composed of interconnected groups of agents defined by their connections, their returns and disposable capital. However, within this framework, the collective states exhibited structural instability, as capital shortages in specific sectors could trigger cascades of defaults.
The present model refines this framework by introducing a third type of agent, banks, a type of investor that can create money through loans. We show that money creation neither eliminates systemic instability nor prevents the emergence of defaults. In fact, the effect of banks on system stability and defaults is ambiguous: When banks favor firms over investors, money creation stabilizes the system by providing the necessary capital to prevent initial defaults, whereas when banks favor investors over firms, investors’ influence is strengthened, potentially amplifying instability and defaults. Moreover, regardless of whether they favor investors or firms, banks may facilitate the propagation of defaults once they have started. Ultimately, because banks are themselves investors, the emergence of highly capitalized, high-return banks can directly generate instability in the system.
Beyond these mechanisms, the analysis reveals the structural limits of macroprudential regulation. Highly capitalized, high-return investors and banks may appear more diversified and resilient, yet they constitute the primary source of endogenous instability. The model thus highlights that systemic fragility is inherent to the very structure of financial interdependence and capital flows.
Key words: Financial Markets, Real Economy, Capital Allocation, Statistical Field Theory, Background fields, Collective states, Multi-Agent Model, Interactions.
JEL Classification: B40, C02, C60, E00, E1, G10
Financial Interactions and Collective States: Investors and Firms
Pierre Gosselin, Aïleen Lotz
Abstract: In a series of papers, we applied a field formalism to analyze capital allocation and accumulation within a microeconomic framework of investors and firms. Financial agents could invest in both firms and other investors, while banks, introduced as investors with a credit multiplier, played a stabilizing or destabilizing role. Two types of interactions were considered within the financial sector: financial agents could either lend capital to or buy shares of other investors. We examined the collective states emerging from these interactions.
At the macro level, we identified multiple collective states, each characterized by distinct levels of average capital and investor distribution across sectors. These states reflect the inherent instability of financial markets, with some configurations leading to default. At the micro level, we analyzed how returns and defaults propagate within a given collective state, highlighting the critical role of banks in stabilizing or amplifying financial fluctuations.
However, these results were derived under the assumption that financial connections were exogenous. The present paper removes this assumption by modeling financial connections as dynamic endogenous variables. Specifically, we extend the framework by introducing a field representation of the network of financial relationships. The collective states previously identified are now embedded in a broader class of states, characterized by the structure of investment shares among investors. We show that these collective states consist of interconnected groups of agents, along with their returns and disposable capital. Depending on the strength and form of connections between agents within each group, collective states may be stable or unstable, allowing for transitions between configurations. In each collective state, some sectors may experience defaults. When the collective state exhibits specific structural conditions, defaults may spread across a significant share of the group.
Key words: Financial Markets, Real Economy, Capital Allocation, Statistical Field Theory, Background fields, Collective states, Multi-Agent Model, Interactions.
JEL Classification: B40, C02, C60, E00, E1, G10
Field Economics: An Introduction
Pierre Gosselin, Aïleen Lotz
Abstract: This paper introduces a novel approach to analyzing economic systems with a large number of agents by applying concepts from field theory. This framework formalizes the probability landscape of the system, capturing both the global collective states and the microeconomic foundations of agent interactions. By identifying the possible collective states of the system, the approach enables a comprehensive understanding of individual dynamics within these states. Furthermore, this perspective challenges traditional economic paradigms, replacing the notion of a single equilibrium with a dynamic view of transitions between collective states.
Key words: Financial Markets, Real Economy, Capital Allocation, Statistical Field Theory, Background fields, Collective states, Multi-Agent Model, Interactions.
JEL Classification: B40, C02, C60, E00, E1, G10
A Statistical Field Perspective on Capital Allocation and Accumulation
Pierre Gosselin, Aïleen Lotz
Abstract: This paper provides a general method to translate a standard economic model with a large number of agents into a field-formalism model. This formalism preserves the system’s interactions and microeconomic features at the individual level but reveals the emergence of collective states.
We apply this method to a simple microeconomic framework of investors and firms. Both macro and micro aspects of the formalism are studied.
At the macro-scale, the field formalism shows that, in each sector, three patterns of capital accumulation may emerge. A distribution of patterns across sectors constitute a collective state. Any change in external parameters or expectations in one sector will affect neighbouring sectors, inducing transitions between collective states and generating permanent fluctuations in patterns and flows of capital. Although changes in expectations can cause abrupt changes in collective states, transitions may be slow to occur. Due to its relative inertia, the real economy is bound to be more affected by these constant variations than the financial markets.
At the micro-scale we compute the transition functions of individual agents and study their probabilistic dynamics in a given collective state, as a function of their initial state. We show that capital accumulation of an individual agent depends on various factors. The probability associated with each firm’s trajectories is the result of several contradictory effects: the firm tends to shift towards sectors with the greatest long-term return, but must take into account the impact of its shift on its attractiveness for investors throughout its trajectory. Since this trajectory depends largely on the average capital of transition sectors, a firm’s attractiveness during its relocation depends on the relative level of capital in those sectors. Moreover, the firm must also consider the effects of competition in the intermediate sectors that tends to oust under-capitalized firm towards sectors with lower average capital. For investors, capital allocation depends on their short and long-term returns and investors will tend to reallocate their capital to maximize both. The higher their level of capital, the stronger the re-allocation will be.
Keywords: Financial Markets, Real Economy, Capital Allocation, Statistical Field Theory, Background fields, Collective states, Multi-Agent Model, Interactions.
JEL Classification: B40, C02, C60, E00, E1, G10.
Statistical Field Theory and Neural Structures Dynamics I, II, III & IV
Pierre Gosselin, Aïleen Lotz
Statistical Field Theory and Neural Structures Dynamics I: Action Functionals, Background States and External Perturbations
Abstract: This series of papers models the dynamics of a large set of interacting neurons within the framework of statistical field theory. The system is described using a two-field model. The first field represents the neuronal activity, while the second field accounts for the interconnections between cells. This model is derived by translating a probabilistic model involving a large number of interacting cells into a field formalism. The current paper focuses on deriving the background fields of the system, which describe the potential equilibria in terms of interconnected groups. Dynamically, we explore the perturbation of these background fields, leading to processes such as activation, association, and reactivation of groups of cells.
Statistical Field Theory and Neural Structures Dynamics II: Signals Propagation, Interferences, Bound States
Abstract: We continue our study of a field formalism for large sets of interacting neurons, together with their connectivity functions. Expanding upon the foundation laid in ([9]), we formulate an effective formalism for the connectivity field in the presence of external sources. We proceed to deduce the propagation of external signals within the system. This enables us to investigate the activation and association of groups of bound cells.
Statistical Field Theory and Neural Structures Dynamics III: Effective Action for Connectivities, Interactions and Emerging Collective States
Abstract: This paper elaborates on the effective field theory for the connectivity field previously introduced in ([7]). We demonstrate that dynamic interactions among connectivities induce modi cations in the background state. These modifications can be understood as the emergence of interacting collective states above the background state. The emergence of such states is contingent on both interactions and the shape of the static or quasi-static background, which acts as a conditioning factor for potential emerging states.
Statistical Field Theory and Neural Structures Dynamics IV: Field-Theoretic Formalism for Interacting Collective States
Abstract: Building upon the findings presented in the first three papers of this series, we formulate an effective field theory for interacting collective states. These states consist of a large number of interconnected neurons and are distinguished by their intrinsic activity. The field theory encompasses an infinite set of fields, each of which characterizes the dynamics of a specific type of collective state. Interaction terms within the theory drive transitions between various collective states, allowing us to describe processes such as activation, association, and deactivation of these states.
I, II, III & IV Keywords: Neural activity, Field theoretic formulation, Transitions, Emerging states, Collective states, connectivity functions.
A Statistical Field Perspective on Capital Allocation and Accumulation Individual dynamics
Pierre Gosselin, Aïleen Lotz
Abstract: We have shown, in a series of articles, that a classical description of a large number of economic agents can be replaced by a statistical fields formalism.
To better understand the accumulation and allocation of capital among different sectors, the present paper applies this statistical fields description to a large number of heterogeneous agents divided into two groups. The first group is composed of a large number of firms in different sectors that collectively own the entire physical capital. The second group, investors, holds the entire financial capital and allocates it between firms across sectors according to investment preferences, expected returns, and stock prices variations on financial markets. In return, firms pay dividends to their investors. Financial capital is thus a function of dividends and stock valuations, whereas physical capital is a function of the total capital allocated by the financial sector. Whereas our previous work focused on the background fields that describe potential long-term equilibria, here we compute the transition functions of individual agents and study their probabilistic dynamics in the background field, as a function of their initial state.
We show that capital accumulation depends on various factors. The probability associated with each firm’s trajectories is the result of several contradictory effects: the firm tends to shift towards sectors with the greatest long-term return, but must take into account the impact of its shift on its attractiveness for investors throughout its trajectory. Since this trajectory depends largely on the average capital of transition sectors, a firm’s attractiveness during its relocation depends on the relative level of capital in those sectors. Thus, an under-capitalized firm reaching a high-capital sector will experience a loss of attractiveness, and subsequently, in investors.
Moreover, the firm must also consider the effects of competition in the intermediate sectors. An under-capitalized firm will tend to be ousted out towards sectors with lower average capital, while an over-capitalized firm will tend to shift towards higher average-capital sectors. For investors, capital allocation depends on their short and long-term returns. These returns are not independent: in the short-term, returns are composed of both the firm’s dividends and the increase in its stock prices. In the long-term, returns are based on the firm’s growth expectations, but also, indirectly, on expectations of higher stock prices. Investors’ capital allocation directly depends on the volatility of stock prices and firms’ dividends. Investors will tend to reallocate their capital to maximize their short and long-term returns. The higher their level of capital, the stronger the re-allocation will be.
Keywords: Financial Markets, Real Economy, Statistical Field Theory, Phase Transition, Capital Allocation, Exchange Space, Multi-Agent Model, Interaction Agents.
JEL Classification: B40, C02, C60, E00, E1, G10
Financial Markets and the Real Economy A Statistical Field Perspective on Capital Allocation and Accumulation
Pierre Gosselin, Aïleen Lotz, Marc Wambst
Abstract: This paper provides a general method to directly translate a classical economic framework with a large number of agents into a field-formalism model. This type of formalism allows the analytical treatment of economic models with an arbitrary number of agents, while preserving the system’s interactions and microeconomic features of the individual level.
We apply this methodology to model the interactions between financial markets and the real economy, described in a classical framework of a large number of heterogeneous agents, investors and firms. Firms are spread among sectors but may shift between sectors to improve their returns. They compete by producing differentiated goods and reward their investors by paying dividends and through their stocks’ valuation. Investors invest in firms and move along sectors based on firms’ expected long-run returns.
The field-formalism model derived from this framework allows for collective states to emerge. We show that the number of firms in each sector depends on the aggregate financial capital invested in the sector and its firms’ expected long-term returns. Capital accumulation in each sector depends both on short-term returns and expected long-term returns relative to neighbouring sectors.
For each sector, three patterns of accumulation emerge. In the first pattern, the dividend component of short-term returns is determinant for sectors with small number of firms and low capital. In the second pattern, both short and long-term returns in the sector drive intermediate-to-high capital. In the third pattern, higher expectations of long-term returns drive massive inputs of capital.
Instability in capital accumulation may arise among and within sectors. We therefore widen our approach and study the dynamics of the collective configurations, in particular interactions between average capital and expected long-term returns, and show that overall stability crucially depends on the expectations’ formation process.
Expectations that are highly reactive to capital variations stabilize high capital configurations, and drive low-to-moderate capital sectors towards zero or a higher level of capital, depending on their initial capital. Inversely, low-to moderate capital configurations are stabilized by expectations moderately reactive to capital variations, and drive high capital sectors towards more moderate level of capital equilibria.
Eventually, the combination of expectations both highly sensitive to exogenous conditions and highly reactive to variations in capital imply that large fluctuations of capital in the system, at the possible expense of the real economy.
Keywords: Financial Markets, Real Economy, Statistical Field Theory, Phase Transition, Capital Allocation, Exchange Space, Multi-Agent Model, Interaction Agents.
JEL Classification: B40, C02, C60, E00, E1, G10
A path integral approach to business cycle models with large number of agents
Pierre Gosselin, Aïleen Lotz, Marc Wambst
Abstract: This paper presents an analytical treatment of economic systems with an arbitrary number of agents that keeps track of the systems’ interactions and agents’ complexity. This formalism does not seek to aggregate agents. It rather replaces the standard optimization approach by a probabilistic description of both the entire system and agents’ behaviors. This is done in two distinct steps. A first step considers an interacting system involving an arbitrary number of agents, where each agent’s utility function is subject to unpredictable shocks. In such a setting, individual optimization problems need not be resolved. Each agent is described by a time-dependent probability distribution centered around his utility optimum. The entire system of agents is thus defined by a composite probability depending on time, agents’ interactions and forward-looking behaviors. This dynamic system is described by a path integral formalism in an abstract space—the space of economic variables—and is very similar to a statistical physics or quantum mechanics system. The usual utility optimization of a representative agent is recovered as a particular case. Compared to a standard optimization, such a description eases the treatment of systems with small number of agents. It becomes however useless for a large number of agents. In a second step therefore, we show that for a large number of agents, the previous description is equivalent to a more compact description in terms of field theory. This yields an analytical though approximate treatment of the system. This field theory does not model the aggregation of a microeconomic system in the usual sense. It rather describes an environment of a large number of interacting agents. From this description, various phases or equilibria may be retrieved, along with individual agents’ behaviors and their interactions with the environment. For illustrative purposes, this paper studies a business cycle model with a large number of agents.
Statistical Field Theory and Networks of Spiking Neurons
Pierre Gosselin, Aïleen Lotz, Marc Wambst
Abstract: This paper models the dynamics of a large set of interacting neurons within the framework of statistical field theory. We use a method initially developed in the context of statistical field theory [44] and later adapted to complex systems in interaction [45][46]. Our model keeps track of individual interacting neurons dynamics but also preserves some of the features and goals of neural field dynamics, such as indexing a large number of neurons by a space variable. Thus, this paper bridges the scale of individual interacting neurons and the macro-scale modelling of neural field theory.